find the generating function to denote the number

Question

Find the generating function to denote the number of ways the sum 9 can be acquired while 2 distinguishable fair dice are tossed and the first shows an even number and the second shows an odd number.

Ans: suppose x₁ and x₂ be the variable denoting the outcomes on the two dice the x₁ has values 2, 4, 6 and x₂ has 1, 3, 5. Here now polynomial for variable x₁ and x₂ can be written as (x² + x⁴ + x⁶) and (x + x³ + x⁵) correspondingly.

So the number of ways sum 9 can be acquired is equal to the coefficient of x⁹ in the expression (x² + x⁴ + x⁶) (x + x³ + x⁵). This is equal to 2.